Calculus Examples

$f (x) = 5 x^{\frac{4}{7}} - x^{\frac{5}{7}}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $5 x^{\frac{4}{7}} - x^{\frac{5}{7}}$ with respect to $x$ is $\frac{d}{d x} [5 x^{\frac{4}{7}}] + \frac{d}{d x} [- x^{\frac{5}{7}}]$ .

$f' (x) = \frac{d}{d x} (5 x^{\frac{4}{7}}) + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2

Evaluate $\frac{d}{d x} [5 x^{\frac{4}{7}}]$ .

Tap for more steps...

Step 1.1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{4}{7}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{4}{7}}]$ .

$f' (x) = 5 \frac{d}{d x} (x^{\frac{4}{7}}) + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{7}$ .

$f' (x) = 5 (\frac{4}{7} x^{\frac{4}{7} - 1}) + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$f' (x) = 5 (\frac{4}{7} x^{\frac{4}{7} - 1 \cdot \frac{7}{7}}) + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2.4

Combine $- 1$ and $\frac{7}{7}$ .

$f' (x) = 5 (\frac{4}{7} x^{\frac{4}{7} + \frac{- 1 \cdot 7}{7}}) + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2.5

Combine the numerators over the common denominator.

$f' (x) = 5 (\frac{4}{7} x^{\frac{4 - 1 \cdot 7}{7}}) + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2.6

Simplify the numerator.

Tap for more steps...

Step 1.1.2.6.1

Multiply $- 1$ by $7$ .

$f' (x) = 5 (\frac{4}{7} x^{\frac{4 - 7}{7}}) + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2.6.2

Subtract $7$ from $4$ .

$f' (x) = 5 (\frac{4}{7} x^{\frac{- 3}{7}}) + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2.7

Move the negative in front of the fraction.

$f' (x) = 5 (\frac{4}{7} x^{- \frac{3}{7}}) + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2.8

Combine $\frac{4}{7}$ and $x^{- \frac{3}{7}}$ .

$f' (x) = 5 (\frac{4 x^{- \frac{3}{7}}}{7}) + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2.9

Combine $5$ and $\frac{4 x^{- \frac{3}{7}}}{7}$ .

$f' (x) = \frac{5 (4 x^{- \frac{3}{7}})}{7} + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2.10

Multiply $4$ by $5$ .

$f' (x) = \frac{20 x^{- \frac{3}{7}}}{7} + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.2.11

Move $x^{- \frac{3}{7}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = \frac{20}{7 x^{\frac{3}{7}}} + \frac{d}{d x} (- x^{\frac{5}{7}})$

Step 1.1.3

Evaluate $\frac{d}{d x} [- x^{\frac{5}{7}}]$ .

Tap for more steps...

Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{5}{7}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{5}{7}}]$ .

$f' (x) = \frac{20}{7 x^{\frac{3}{7}}} - \frac{d}{d x} x^{\frac{5}{7}}$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{7}$ .

$f' (x) = \frac{20}{7 x^{\frac{3}{7}}} - (\frac{5}{7} x^{\frac{5}{7} - 1})$

Step 1.1.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$f' (x) = \frac{20}{7 x^{\frac{3}{7}}} - (\frac{5}{7} x^{\frac{5}{7} - 1 \cdot \frac{7}{7}})$

Step 1.1.3.4

Combine $- 1$ and $\frac{7}{7}$ .

$f' (x) = \frac{20}{7 x^{\frac{3}{7}}} - (\frac{5}{7} x^{\frac{5}{7} + \frac{- 1 \cdot 7}{7}})$

Step 1.1.3.5

Combine the numerators over the common denominator.

$f' (x) = \frac{20}{7 x^{\frac{3}{7}}} - (\frac{5}{7} x^{\frac{5 - 1 \cdot 7}{7}})$

Step 1.1.3.6

Simplify the numerator.

Tap for more steps...

Step 1.1.3.6.1

Multiply $- 1$ by $7$ .

$f' (x) = \frac{20}{7 x^{\frac{3}{7}}} - (\frac{5}{7} x^{\frac{5 - 7}{7}})$

Step 1.1.3.6.2

Subtract $7$ from $5$ .

$f' (x) = \frac{20}{7 x^{\frac{3}{7}}} - (\frac{5}{7} x^{\frac{- 2}{7}})$

Step 1.1.3.7

Move the negative in front of the fraction.

$f' (x) = \frac{20}{7 x^{\frac{3}{7}}} - (\frac{5}{7} x^{- \frac{2}{7}})$

Step 1.1.3.8

Combine $\frac{5}{7}$ and $x^{- \frac{2}{7}}$ .

$f' (x) = \frac{20}{7 x^{\frac{3}{7}}} - \frac{5 x^{- \frac{2}{7}}}{7}$

Step 1.1.3.9

Move $x^{- \frac{2}{7}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = \frac{20}{7 x^{\frac{3}{7}}} - \frac{5}{7 x^{\frac{2}{7}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{20}{7 x^{\frac{3}{7}}} - \frac{5}{7 x^{\frac{2}{7}}}$ .

$\frac{20}{7 x^{\frac{3}{7}}} - \frac{5}{7 x^{\frac{2}{7}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{20}{7 x^{\frac{3}{7}}} - \frac{5}{7 x^{\frac{2}{7}}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$\frac{20}{7 x^{\frac{3}{7}}} - \frac{5}{7 x^{\frac{2}{7}}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$7 x^{\frac{3}{7}}, 7 x^{\frac{2}{7}}, 1$

Step 2.2.2

Since $7 x^{\frac{3}{7}}, 7 x^{\frac{2}{7}}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $7, 7, 1$ then find LCM for the variable part $x^{\frac{3}{7}}, x^{\frac{2}{7}}$ .

Step 2.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.2.4

Since $7$ has no factors besides $1$ and $7$ .

$7$ is a prime number

Step 2.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.2.6

The LCM of $7, 7, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$7$

Step 2.2.7

The LCM of $x^{\frac{3}{7}}, x^{\frac{2}{7}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x^{\frac{3}{7}}$

Step 2.2.8

The LCM for $7 x^{\frac{3}{7}}, 7 x^{\frac{2}{7}}, 1$ is the numeric part $7$ multiplied by the variable part.

$7 x^{\frac{3}{7}}$

Step 2.3

Multiply each term in $\frac{20}{7 x^{\frac{3}{7}}} - \frac{5}{7 x^{\frac{2}{7}}} = 0$ by $7 x^{\frac{3}{7}}$ to eliminate the fractions.

Tap for more steps...

Step 2.3.1

Multiply each term in $\frac{20}{7 x^{\frac{3}{7}}} - \frac{5}{7 x^{\frac{2}{7}}} = 0$ by $7 x^{\frac{3}{7}}$ .

$\frac{20}{7 x^{\frac{3}{7}}} (7 x^{\frac{3}{7}}) - \frac{5}{7 x^{\frac{2}{7}}} (7 x^{\frac{3}{7}}) = 0 (7 x^{\frac{3}{7}})$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Simplify each term.

Tap for more steps...

Step 2.3.2.1.1

Rewrite using the commutative property of multiplication.

$7 \frac{20}{7 x^{\frac{3}{7}}} x^{\frac{3}{7}} - \frac{5}{7 x^{\frac{2}{7}}} (7 x^{\frac{3}{7}}) = 0 (7 x^{\frac{3}{7}})$

Step 2.3.2.1.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 2.3.2.1.2.1

Cancel the common factor.

$7 \frac{20}{7 x^{\frac{3}{7}}} x^{\frac{3}{7}} - \frac{5}{7 x^{\frac{2}{7}}} (7 x^{\frac{3}{7}}) = 0 (7 x^{\frac{3}{7}})$

Step 2.3.2.1.2.2

Rewrite the expression.

$\frac{20}{x^{\frac{3}{7}}} x^{\frac{3}{7}} - \frac{5}{7 x^{\frac{2}{7}}} (7 x^{\frac{3}{7}}) = 0 (7 x^{\frac{3}{7}})$

Step 2.3.2.1.3

Cancel the common factor of $x^{\frac{3}{7}}$ .

Tap for more steps...

Step 2.3.2.1.3.1

Cancel the common factor.

$\frac{20}{x^{\frac{3}{7}}} x^{\frac{3}{7}} - \frac{5}{7 x^{\frac{2}{7}}} (7 x^{\frac{3}{7}}) = 0 (7 x^{\frac{3}{7}})$

Step 2.3.2.1.3.2

Rewrite the expression.

$20 - \frac{5}{7 x^{\frac{2}{7}}} (7 x^{\frac{3}{7}}) = 0 (7 x^{\frac{3}{7}})$

Step 2.3.2.1.4

Cancel the common factor of $x^{\frac{2}{7}} \cdot 7$ .

Tap for more steps...

Step 2.3.2.1.4.1

Move the leading negative in $- \frac{5}{7 x^{\frac{2}{7}}}$ into the numerator.

$20 + \frac{- 5}{7 x^{\frac{2}{7}}} (7 x^{\frac{3}{7}}) = 0 (7 x^{\frac{3}{7}})$

Step 2.3.2.1.4.2

Factor $x^{\frac{2}{7}} \cdot 7$ out of $7 x^{\frac{2}{7}}$ .

$20 + \frac{- 5}{x^{\frac{2}{7}} \cdot 7 (1)} (7 x^{\frac{3}{7}}) = 0 (7 x^{\frac{3}{7}})$

Step 2.3.2.1.4.3

Factor $x^{\frac{2}{7}} \cdot 7$ out of $7 x^{\frac{3}{7}}$ .

$20 + \frac{- 5}{x^{\frac{2}{7}} \cdot 7 (1)} (x^{\frac{2}{7}} \cdot 7 (x^{\frac{1}{7}})) = 0 (7 x^{\frac{3}{7}})$

Step 2.3.2.1.4.4

Cancel the common factor.

$20 + \frac{- 5}{x^{\frac{2}{7}} \cdot 7 \cdot 1} (x^{\frac{2}{7}} \cdot 7 x^{\frac{1}{7}}) = 0 (7 x^{\frac{3}{7}})$

Step 2.3.2.1.4.5

Rewrite the expression.

$20 - 5 x^{\frac{1}{7}} = 0 (7 x^{\frac{3}{7}})$

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Multiply $0 (7 x^{\frac{3}{7}})$ .

Tap for more steps...

Step 2.3.3.1.1

Multiply $7$ by $0$ .

$20 - 5 x^{\frac{1}{7}} = 0 x^{\frac{3}{7}}$

Step 2.3.3.1.2

Multiply $0$ by $x^{\frac{3}{7}}$ .

$20 - 5 x^{\frac{1}{7}} = 0$

Step 2.4

Solve the equation.

Tap for more steps...

Step 2.4.1

Subtract $20$ from both sides of the equation.

$- 5 x^{\frac{1}{7}} = - 20$

Step 2.4.2

Raise each side of the equation to the power of $7$ to eliminate the fractional exponent on the left side.

${(- 5 x^{\frac{1}{7}})}^{7} = {(- 20)}^{7}$

Step 2.4.3

Simplify the exponent.

Tap for more steps...

Step 2.4.3.1

Simplify the left side.

Tap for more steps...

Step 2.4.3.1.1

Simplify ${(- 5 x^{\frac{1}{7}})}^{7}$ .

Tap for more steps...

Step 2.4.3.1.1.1

Apply the product rule to $- 5 x^{\frac{1}{7}}$ .

${(- 5)}^{7} {(x^{\frac{1}{7}})}^{7} = {(- 20)}^{7}$

Step 2.4.3.1.1.2

Raise $- 5$ to the power of $7$ .

$- 78125 {(x^{\frac{1}{7}})}^{7} = {(- 20)}^{7}$

Step 2.4.3.1.1.3

Multiply the exponents in ${(x^{\frac{1}{7}})}^{7}$ .

Tap for more steps...

Step 2.4.3.1.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 78125 x^{\frac{1}{7} \cdot 7} = {(- 20)}^{7}$

Step 2.4.3.1.1.3.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 2.4.3.1.1.3.2.1

Cancel the common factor.

$- 78125 x^{\frac{1}{7} \cdot 7} = {(- 20)}^{7}$

Step 2.4.3.1.1.3.2.2

Rewrite the expression.

$- 78125 x^{1} = {(- 20)}^{7}$

Step 2.4.3.1.1.4

Simplify.

$- 78125 x = {(- 20)}^{7}$

Step 2.4.3.2

Simplify the right side.

Tap for more steps...

Step 2.4.3.2.1

Raise $- 20$ to the power of $7$ .

$- 78125 x = - 1280000000$

Step 2.4.4

Divide each term in $- 78125 x = - 1280000000$ by $- 78125$ and simplify.

Tap for more steps...

Step 2.4.4.1

Divide each term in $- 78125 x = - 1280000000$ by $- 78125$ .

$\frac{- 78125 x}{- 78125} = \frac{- 1280000000}{- 78125}$

Step 2.4.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.4.2.1

Cancel the common factor of $- 78125$ .

Tap for more steps...

Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{- 78125 x}{- 78125} = \frac{- 1280000000}{- 78125}$

Step 2.4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1280000000}{- 78125}$

Step 2.4.4.3

Simplify the right side.

Tap for more steps...

Step 2.4.4.3.1

Divide $- 1280000000$ by $- 78125$ .

$x = 16384$

Step 3

The values which make the derivative equal to $0$ are $16384$ .

$16384$

Step 4

Find where the derivative is undefined.

Tap for more steps...

Step 4.1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 4.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{20}{7 \sqrt[7]{x^{3}}} - \frac{5}{7 x^{\frac{2}{7}}}$

Step 4.1.2

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{20}{7 \sqrt[7]{x^{3}}} - \frac{5}{7 \sqrt[7]{x^{2}}}$

Step 4.2

Set the denominator in $\frac{20}{7 \sqrt[7]{x^{3}}}$ equal to $0$ to find where the expression is undefined.

$7 \sqrt[7]{x^{3}} = 0$

Step 4.3

Solve for $x$ .

Tap for more steps...

Step 4.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $7$ .

${(7 \sqrt[7]{x^{3}})}^{7} = 0^{7}$

Step 4.3.2

Simplify each side of the equation.

Tap for more steps...

Step 4.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{3}}$ as $x^{\frac{3}{7}}$ .

${(7 x^{\frac{3}{7}})}^{7} = 0^{7}$

Step 4.3.2.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.2.1

Simplify ${(7 x^{\frac{3}{7}})}^{7}$ .

Tap for more steps...

Step 4.3.2.2.1.1

Apply the product rule to $7 x^{\frac{3}{7}}$ .

$7^{7} {(x^{\frac{3}{7}})}^{7} = 0^{7}$

Step 4.3.2.2.1.2

Raise $7$ to the power of $7$ .

$823543 {(x^{\frac{3}{7}})}^{7} = 0^{7}$

Step 4.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{3}{7}})}^{7}$ .

Tap for more steps...

Step 4.3.2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$823543 x^{\frac{3}{7} \cdot 7} = 0^{7}$

Step 4.3.2.2.1.3.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.3.2.2.1.3.2.1

Cancel the common factor.

$823543 x^{\frac{3}{7} \cdot 7} = 0^{7}$

Step 4.3.2.2.1.3.2.2

Rewrite the expression.

$823543 x^{3} = 0^{7}$

Step 4.3.2.3

Simplify the right side.

Tap for more steps...

Step 4.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$823543 x^{3} = 0$

Step 4.3.3

Solve for $x$ .

Tap for more steps...

Step 4.3.3.1

Divide each term in $823543 x^{3} = 0$ by $823543$ and simplify.

Tap for more steps...

Step 4.3.3.1.1

Divide each term in $823543 x^{3} = 0$ by $823543$ .

$\frac{823543 x^{3}}{823543} = \frac{0}{823543}$

Step 4.3.3.1.2

Simplify the left side.

Tap for more steps...

Step 4.3.3.1.2.1

Cancel the common factor of $823543$ .

Tap for more steps...

Step 4.3.3.1.2.1.1

Cancel the common factor.

$\frac{823543 x^{3}}{823543} = \frac{0}{823543}$

Step 4.3.3.1.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{0}{823543}$

Step 4.3.3.1.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1.3.1

Divide $0$ by $823543$ .

$x^{3} = 0$

Step 4.3.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 4.3.3.3

Simplify $\sqrt[3]{0}$ .

Tap for more steps...

Step 4.3.3.3.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 4.3.3.3.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 4.4

Set the denominator in $\frac{5}{7 \sqrt[7]{x^{2}}}$ equal to $0$ to find where the expression is undefined.

$7 \sqrt[7]{x^{2}} = 0$

Step 4.5

Solve for $x$ .

Tap for more steps...

Step 4.5.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $7$ .

${(7 \sqrt[7]{x^{2}})}^{7} = 0^{7}$

Step 4.5.2

Simplify each side of the equation.

Tap for more steps...

Step 4.5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{2}}$ as $x^{\frac{2}{7}}$ .

${(7 x^{\frac{2}{7}})}^{7} = 0^{7}$

Step 4.5.2.2

Simplify the left side.

Tap for more steps...

Step 4.5.2.2.1

Simplify ${(7 x^{\frac{2}{7}})}^{7}$ .

Tap for more steps...

Step 4.5.2.2.1.1

Apply the product rule to $7 x^{\frac{2}{7}}$ .

$7^{7} {(x^{\frac{2}{7}})}^{7} = 0^{7}$

Step 4.5.2.2.1.2

Raise $7$ to the power of $7$ .

$823543 {(x^{\frac{2}{7}})}^{7} = 0^{7}$

Step 4.5.2.2.1.3

Multiply the exponents in ${(x^{\frac{2}{7}})}^{7}$ .

Tap for more steps...

Step 4.5.2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$823543 x^{\frac{2}{7} \cdot 7} = 0^{7}$

Step 4.5.2.2.1.3.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.5.2.2.1.3.2.1

Cancel the common factor.

$823543 x^{\frac{2}{7} \cdot 7} = 0^{7}$

Step 4.5.2.2.1.3.2.2

Rewrite the expression.

$823543 x^{2} = 0^{7}$

Step 4.5.2.3

Simplify the right side.

Tap for more steps...

Step 4.5.2.3.1

Raising $0$ to any positive power yields $0$ .

$823543 x^{2} = 0$

Step 4.5.3

Solve for $x$ .

Tap for more steps...

Step 4.5.3.1

Divide each term in $823543 x^{2} = 0$ by $823543$ and simplify.

Tap for more steps...

Step 4.5.3.1.1

Divide each term in $823543 x^{2} = 0$ by $823543$ .

$\frac{823543 x^{2}}{823543} = \frac{0}{823543}$

Step 4.5.3.1.2

Simplify the left side.

Tap for more steps...

Step 4.5.3.1.2.1

Cancel the common factor of $823543$ .

Tap for more steps...

Step 4.5.3.1.2.1.1

Cancel the common factor.

$\frac{823543 x^{2}}{823543} = \frac{0}{823543}$

Step 4.5.3.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{823543}$

Step 4.5.3.1.3

Simplify the right side.

Tap for more steps...

Step 4.5.3.1.3.1

Divide $0$ by $823543$ .

$x^{2} = 0$

Step 4.5.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 4.5.3.3

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 4.5.3.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 4.5.3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 4.5.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, 0) \cup (0, 16384) \cup (16384, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = \frac{20}{7 {(- 1)}^{\frac{3}{7}}} - \frac{5}{7 {(- 1)}^{\frac{2}{7}}}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Simplify the denominator.

Tap for more steps...

Step 6.2.1.1.1

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$f' (- 1) = \frac{20}{7 {({(- 1)}^{7})}^{\frac{3}{7}}} - \frac{5}{7 {(- 1)}^{\frac{2}{7}}}$

Step 6.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f' (- 1) = \frac{20}{7 {(- 1)}^{7 (\frac{3}{7})}} - \frac{5}{7 {(- 1)}^{\frac{2}{7}}}$

Step 6.2.1.1.3

Cancel the common factor of $7$ .

Tap for more steps...

Step 6.2.1.1.3.1

Cancel the common factor.

$f' (- 1) = \frac{20}{7 {(- 1)}^{7 (\frac{3}{7})}} - \frac{5}{7 {(- 1)}^{\frac{2}{7}}}$

Step 6.2.1.1.3.2

Rewrite the expression.

$f' (- 1) = \frac{20}{7 {(- 1)}^{3}} - \frac{5}{7 {(- 1)}^{\frac{2}{7}}}$

Step 6.2.1.1.4

Raise $- 1$ to the power of $3$ .

$f' (- 1) = \frac{20}{7 \cdot - 1} - \frac{5}{7 {(- 1)}^{\frac{2}{7}}}$

Step 6.2.1.2

Multiply $7$ by $- 1$ .

$f' (- 1) = \frac{20}{- 7} - \frac{5}{7 {(- 1)}^{\frac{2}{7}}}$

Step 6.2.1.3

Move the negative in front of the fraction.

$f' (- 1) = - \frac{20}{7} - \frac{5}{7 {(- 1)}^{\frac{2}{7}}}$

Step 6.2.1.4

Simplify the denominator.

Tap for more steps...

Step 6.2.1.4.1

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$f' (- 1) = - \frac{20}{7} - \frac{5}{7 {({(- 1)}^{7})}^{\frac{2}{7}}}$

Step 6.2.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f' (- 1) = - \frac{20}{7} - \frac{5}{7 {(- 1)}^{7 (\frac{2}{7})}}$

Step 6.2.1.4.3

Cancel the common factor of $7$ .

Tap for more steps...

Step 6.2.1.4.3.1

Cancel the common factor.

$f' (- 1) = - \frac{20}{7} - \frac{5}{7 {(- 1)}^{7 (\frac{2}{7})}}$

Step 6.2.1.4.3.2

Rewrite the expression.

$f' (- 1) = - \frac{20}{7} - \frac{5}{7 {(- 1)}^{2}}$

Step 6.2.1.4.4

Raise $- 1$ to the power of $2$ .

$f' (- 1) = - \frac{20}{7} - \frac{5}{7 \cdot 1}$

Step 6.2.1.5

Multiply $7$ by $1$ .

$f' (- 1) = - \frac{20}{7} - \frac{5}{7}$

Step 6.2.2

Combine fractions.

Tap for more steps...

Step 6.2.2.1

Combine the numerators over the common denominator.

$f' (- 1) = \frac{- 20 - 5}{7}$

Step 6.2.2.2

Simplify the expression.

Tap for more steps...

Step 6.2.2.2.1

Subtract $5$ from $- 20$ .

$f' (- 1) = \frac{- 25}{7}$

Step 6.2.2.2.2

Move the negative in front of the fraction.

$f' (- 1) = - \frac{25}{7}$

Step 6.2.3

The final answer is $- \frac{25}{7}$ .

$- \frac{25}{7}$

Step 6.3

At $x = - 1$ the derivative is $- \frac{25}{7}$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

Decreasing on $(- \infty, 0)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(0, 16384)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 7.1

Replace the variable $x$ with $8192$ in the expression.

$f' (8192) = \frac{20}{7 {(8192)}^{\frac{3}{7}}} - \frac{5}{7 {(8192)}^{\frac{2}{7}}}$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Remove parentheses.

$f' (8192) = \frac{20}{7 \cdot 8192^{\frac{3}{7}}} - \frac{5}{7 \cdot 8192^{\frac{2}{7}}}$

Step 7.2.2

To write $- \frac{5}{7 \cdot 8192^{\frac{2}{7}}}$ as a fraction with a common denominator, multiply by $\frac{8192^{\frac{1}{7}}}{8192^{\frac{1}{7}}}$ .

$f' (8192) = \frac{20}{7 \cdot 8192^{\frac{3}{7}}} - \frac{5}{7 \cdot 8192^{\frac{2}{7}}} \cdot \frac{8192^{\frac{1}{7}}}{8192^{\frac{1}{7}}}$

Step 7.2.3

Write each expression with a common denominator of $7 \cdot 8192^{\frac{3}{7}}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 7.2.3.1

Multiply $\frac{5}{7 \cdot 8192^{\frac{2}{7}}}$ by $\frac{8192^{\frac{1}{7}}}{8192^{\frac{1}{7}}}$ .

$f' (8192) = \frac{20}{7 \cdot 8192^{\frac{3}{7}}} - \frac{5 \cdot 8192^{\frac{1}{7}}}{7 \cdot 8192^{\frac{2}{7}} \cdot 8192^{\frac{1}{7}}}$

Step 7.2.3.2

Multiply $8192^{\frac{2}{7}}$ by $8192^{\frac{1}{7}}$ by adding the exponents.

Tap for more steps...

Step 7.2.3.2.1

Move $8192^{\frac{1}{7}}$ .

$f' (8192) = \frac{20}{7 \cdot 8192^{\frac{3}{7}}} - \frac{5 \cdot 8192^{\frac{1}{7}}}{7 \cdot (8192^{\frac{1}{7}} \cdot 8192^{\frac{2}{7}})}$

Step 7.2.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (8192) = \frac{20}{7 \cdot 8192^{\frac{3}{7}}} - \frac{5 \cdot 8192^{\frac{1}{7}}}{7 \cdot 8192^{\frac{1}{7} + \frac{2}{7}}}$

Step 7.2.3.2.3

Combine the numerators over the common denominator.

$f' (8192) = \frac{20}{7 \cdot 8192^{\frac{3}{7}}} - \frac{5 \cdot 8192^{\frac{1}{7}}}{7 \cdot 8192^{\frac{1 + 2}{7}}}$

Step 7.2.3.2.4

Add $1$ and $2$ .

$f' (8192) = \frac{20}{7 \cdot 8192^{\frac{3}{7}}} - \frac{5 \cdot 8192^{\frac{1}{7}}}{7 \cdot 8192^{\frac{3}{7}}}$

Step 7.2.4

Combine the numerators over the common denominator.

$f' (8192) = \frac{20 - 5 \cdot 8192^{\frac{1}{7}}}{7 \cdot 8192^{\frac{3}{7}}}$

Step 7.2.5

The final answer is $\frac{20 - 5 \cdot 8192^{\frac{1}{7}}}{7 \cdot 8192^{\frac{3}{7}}}$ .

$\frac{20 - 5 \cdot 8192^{\frac{1}{7}}}{7 \cdot 8192^{\frac{3}{7}}}$

Step 7.3

At $x = 8192$ the derivative is $\frac{20 - 5 \cdot 8192^{\frac{1}{7}}}{7 \cdot 8192^{\frac{3}{7}}}$ . Since this is negative, the function is decreasing on $(0, 16384)$ .

Decreasing on $(0, 16384)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(16384, \infty)$ into the derivative to determine if the function is increasing or decreasing.

Tap for more steps...

Step 8.1

Replace the variable $x$ with $16385$ in the expression.

$f' (16385) = \frac{20}{7 {(16385)}^{\frac{3}{7}}} - \frac{5}{7 {(16385)}^{\frac{2}{7}}}$

Step 8.2

Simplify the result.

Tap for more steps...

Step 8.2.1

Remove parentheses.

$f' (16385) = \frac{20}{7 \cdot 16385^{\frac{3}{7}}} - \frac{5}{7 \cdot 16385^{\frac{2}{7}}}$

Step 8.2.2

To write $- \frac{5}{7 \cdot 16385^{\frac{2}{7}}}$ as a fraction with a common denominator, multiply by $\frac{16385^{\frac{1}{7}}}{16385^{\frac{1}{7}}}$ .

$f' (16385) = \frac{20}{7 \cdot 16385^{\frac{3}{7}}} - \frac{5}{7 \cdot 16385^{\frac{2}{7}}} \cdot \frac{16385^{\frac{1}{7}}}{16385^{\frac{1}{7}}}$

Step 8.2.3

Write each expression with a common denominator of $7 \cdot 16385^{\frac{3}{7}}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 8.2.3.1

Multiply $\frac{5}{7 \cdot 16385^{\frac{2}{7}}}$ by $\frac{16385^{\frac{1}{7}}}{16385^{\frac{1}{7}}}$ .

$f' (16385) = \frac{20}{7 \cdot 16385^{\frac{3}{7}}} - \frac{5 \cdot 16385^{\frac{1}{7}}}{7 \cdot 16385^{\frac{2}{7}} \cdot 16385^{\frac{1}{7}}}$

Step 8.2.3.2

Multiply $16385^{\frac{2}{7}}$ by $16385^{\frac{1}{7}}$ by adding the exponents.

Tap for more steps...

Step 8.2.3.2.1

Move $16385^{\frac{1}{7}}$ .

$f' (16385) = \frac{20}{7 \cdot 16385^{\frac{3}{7}}} - \frac{5 \cdot 16385^{\frac{1}{7}}}{7 \cdot (16385^{\frac{1}{7}} \cdot 16385^{\frac{2}{7}})}$

Step 8.2.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (16385) = \frac{20}{7 \cdot 16385^{\frac{3}{7}}} - \frac{5 \cdot 16385^{\frac{1}{7}}}{7 \cdot 16385^{\frac{1}{7} + \frac{2}{7}}}$

Step 8.2.3.2.3

Combine the numerators over the common denominator.

$f' (16385) = \frac{20}{7 \cdot 16385^{\frac{3}{7}}} - \frac{5 \cdot 16385^{\frac{1}{7}}}{7 \cdot 16385^{\frac{1 + 2}{7}}}$

Step 8.2.3.2.4

Add $1$ and $2$ .

$f' (16385) = \frac{20}{7 \cdot 16385^{\frac{3}{7}}} - \frac{5 \cdot 16385^{\frac{1}{7}}}{7 \cdot 16385^{\frac{3}{7}}}$

Step 8.2.4

Combine the numerators over the common denominator.

$f' (16385) = \frac{20 - 5 \cdot 16385^{\frac{1}{7}}}{7 \cdot 16385^{\frac{3}{7}}}$

Step 8.2.5

The final answer is $\frac{20 - 5 \cdot 16385^{\frac{1}{7}}}{7 \cdot 16385^{\frac{3}{7}}}$ .

$\frac{20 - 5 \cdot 16385^{\frac{1}{7}}}{7 \cdot 16385^{\frac{3}{7}}}$

Step 8.3

At $x = 16385$ the derivative is $\frac{20 - 5 \cdot 16385^{\frac{1}{7}}}{7 \cdot 16385^{\frac{3}{7}}}$ . Since this is negative, the function is decreasing on $(16384, \infty)$ .

Decreasing on $(16384, \infty)$ since $f' (x) < 0$

Step 9

List the intervals on which the function is increasing and decreasing.

Decreasing on: $(- \infty, 0), (0, 16384), (16384, \infty)$

Step 10